Elsevier

Neural Networks

Volume 149, May 2022, Pages 137-145
Neural Networks

Finite-time and sampled-data synchronization of complex dynamical networks subject to average dwell-time switching signal

https://doi.org/10.1016/j.neunet.2022.02.013Get rights and content

Abstract

This study deals with the finite-time synchronization problem of a class of switched complex dynamical networks (CDNs) with distributed coupling delays via sampled-data control. First, the dynamical model is studied with coupling delays in more detail. The sampling system is then converted to a continuous time-delay system using an input delay technique. We obtain some unique and less conservative criteria on exponential stability using the Lyapunov–Krasovskii functional (LKF), which is generated with a Kronecker product, linear matrix inequalities (LMIs), and integral inequality. Furthermore, some sufficient criteria are derived by an average dwell-time method and determine the finite-time boundedness of CDNs with switching signal. The proposed sufficient conditions can be represented in the form of LMIs. Finally, numerical examples are given to show that the suggested strategy is feasible.

Introduction

Coupled neural networks, as particular problems of complex networks, have gotten a lot of interest and have been extensively investigated for their possible applications in a variety of fields. Set squares are used to quantify linked nodes in complex dynamical networks (CDNs), with each node corresponding to a dynamical network. Complex dynamical networks are widely shown as a wide range of observable systems, including electrical distribution networks, social networks, the World Wide Web (WWW), scientific citation networks, and etc., (see Boccaletti et al., 2006, Strogatz, 2001, Wang and Chen, 2003 and the references therein). In actuality, the phenomenon of synchronization happens frequently in nature. The synchronization issue for all dynamical nodes in the network is studied when the dynamical networks are first characterized in terms of a differential equation with a coupling term between dynamical nodes in Divya, Sakthivel, and Liu (2021) and Wang and Chen (2002). For several kinds of CDNs, synchronization requirements have also been proposed in Chu et al., 2020, Fang, 2015, Kaviarasan et al., 2020, Li et al., 2018, Lu et al., 2020, Sakthivel et al., 2021, Wang et al., 2017 and Xing, Lu, Qiu, and Shen (2021).

Recently, topology and node switching must be considered when complex networks are influenced by jumping parameters, link failure and new link construction, and other environmental changes in the real world. In light of the switching phenomena that occur in neural networks (NNs), certain switched NNs models, including coupled switched NNs have been proposed, and a number of helpful research on the dynamical analysis of switched NNs have been given. For example, the stability and passivity of switched NNs with time-varying delays were studied in Lian and Wang (2015) and Sakthivel, Sakthivel, Kaviarasan, Wang, and Ma (2018), and discrete-time switched NNs and stochastic switched NNs were investigated in Wang, Jiang, Hu, and Ma (2021) and Ye, Ji, and Zhang (2016). Neural networks are large-scale nonlinear systems that are made up of numerous subsystems. Existing linkages between neurons may be severed and new ones created as subsystems transition. This would drastically alter the system’s connectivity topology. The so-called switched NNs are frequently investigated to gain a thorough and clear knowledge of the dynamics of these complex systems. Each switched NNs are made up of a set of continuous-time or discrete-time subsystems, with a rule that governs the switching between them (Huang et al., 2016, Wang et al., 2019). In most extant literature’s, the common Lyapunov functional (CLF) technique and the multiple Lyapunov functional (MLF) approach are the two most often used methods for synthesizing the related findings. However, for switched networks with arbitrary switching signals, finding a CLF is difficult. MLF methods can be utilized successfully with switching signal limitations.

Synchronization is one of the most essential dynamical features of complex networks, and it has long been a hot study area. Synchronization of CDNs is a critical collective dynamical characteristic that has garnered a lot of attention. Synchronization was explored in relation to linked NNs with delays (Lu & Chen, 2004). Global synchronization criteria of CDNs scheme was developed in Cao, Li, and Wang (2006) using the Lyapunov functional technique and linear matrix inequalities (LMIs). The synchronization findings may be obtained in Gunasekaran, Saravanakumar, Joo, and Kim (2019) and Kaviarasan et al. (2020), when complex networks with coupling weights are studied. To date, many types of synchronization and numerous control systems have been presented, such as NNs with delay coupling through impulsive control (He, Qian, & Cao, 2017), Adaptive control, and parameter identification are used in the development of neural networks (NNs) in Zhou, Chen, and Xiang (2006), the use of multiagent networks to linked circuits via sampled-data control (Gunasekaran, Zhai, & Yu, 2020). In summary, the majority of the synchronization findings are defined at infinite time intervals and are based on Lyapunov stability theory. In many real-world applications, synchronous behavior at finite-time intervals may be more essential. Several finite-time synchronization solutions for complex networks have recently been suggested (See Ali et al., 2018, Huang et al., 2020, Qiu et al., 2018, Yang et al., 2020). As the name implies, finite-time synchronization occurs when a CDNs achieves synchronization in a finite amount of time. This is highly beneficial in the realms of instant messaging and certain practical engineering.

As far as we know, the majority of synchronization control techniques for CDNs are based on the synchronous switching assumption, which states that each subnetwork’s controller can perfectly match the switching signal of the actual network. In fact, due to the unknown switching signal of networks, it is difficult to implement since it takes some time to detect the switching signal and active subnetworks, and then select the appropriate controller. In addition, control strategies to improve the control performance of dynamic systems require digital feedback. This has the advantages of increased speed, increased reliability, smaller size, efficiency, ease of installation and maintenance, and lower cost when implementing a continuous-time system. In addition, sampled-data control is gaining popularity, with the goal of making the considered system stable with appropriate performance and good control performance. There are a lot of interesting results for sampled-data synchronization of different dynamical frameworks (See Ali et al., 2017, Ali et al., 2019, Anbuvithya et al., 2021, Dharani et al., 2017, Ding and Zhu, 2020, Gunasekaran et al., 2021, Lee and Park, 2017, Theesar et al., 2012, Vadivel et al., 2021, Wang, Ding, and Li, 2021). Nevertheless, there are still no advances made in the guaranteed finite-time and sampled-data synchronization of switched complex dynamical networks (SCDNs).

In this study, we look at the topic of finite-time synchronization analysis of switched CDNs with coupling delay using sampled-data control, which is motivated by the previous discussion. The sufficient conditions are derived using the Lyapunov functional approach and Kronecker product methods. Then, new sufficient conditions are required for finite-time synchronization of such network models by generating proper Lyapunov functionals and employing the free weight matrix technique and the LMIs. Finally, numerical examples are presented to demonstrate the applicability and efficacy of the suggested technique.

Section snippets

Problem statement and preliminaries

Consider the CDNs model, which has N identical connected nodes: ẋi(t)=f(xi(t))+ς1j=1NGij(1)Axj(tτ(t))+ς2j=1NGij(2)Btτtxj(s)ds+uĩ(t),x(t)=φ(t),t[t0τ,t0],τ>0,i=1toN,where xi(t)Rn is the state vector of the ith network at time t. uĩ(t) are the control inputs. f:RnRn is a continuous vector-valued function. ς1>0 and ς2>0 represent the coupling strengths. A={aij}n×nRn×n,B={bij}n×nRn×n are constant inner-coupling matrices of the nodes, and Gk=(Gij(k))N×N,k=1,2, is the outer-coupling

Main results

This section shows the conditions for finite-time synchronization of SCDNs (8) using sampled-data controller.

Theorem 1

Under Assumption (A), for given gains Kk, positive scalars τ,h,ς1,ς2,αk and μ, the error SCDNs (8) is finite-time synchronized with respect to (c1,c2,h,T,R) if there exist matrices Pk>0,Q1k>0,Q2k>0,Rk>0 and scalar γ>0, and matrices Jk with appropriate dimensions such that the following LMIs hold for any kN [Ωij]k7×7<0, ϖ2c1<ϖ1c2eα¯T,and the average dwell time Ta of the switching

Numerical examples

We provide numerical examples in this part to demonstrate the efficacy of the suggested technique.

Example 4.1

Consider the following SCDNs (1): A1=1001,A2=0.8000.7,B1=0.8000.9,B2=0.6000.4.

Consider the outer coupling matrices as G1i and G2i with G11=G12=101011112,G21=G22=101021111. The function f is taken as f(xi(t))=0.5xi1+tanh(0.2xi1)+0.2xi10.95xi2tanh(0.75xi2),which results in U¯=0.50.200.95,V¯=0.30.200.2.

By employing Theorem 2, and the lower bound value Ta of the ADT on the signal σ can be

Conclusion

In this paper, we studied how to design SCDNs finite-time synchronization with hybrid coupling through sampled-data control. A new synchronization criterion was obtained using the Lyapunov method and the Kronecker product method, based on the appropriate LKF with the appropriate integral term. In addition, the sampled-data controller is designed to synchronize the SCDNs for a finite amount of time. At the same time, by solving the LMIs, the desired controller was obtained. Finally, two

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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